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Simplifying 414.48 = 2 * 3.14r(r) + 2 * 3.14r * 5.5 Multiply 2 * 3.14 414.48 = 6.28r * r + 2 * 3.14r * 5.5 Multiply r * r 414.48 = 6.28r2 + 2 * 3.14r * 5.5 Reorder the terms for easier multiplication: 414.48 = 6.28r2 + 2 * 3.14 * 5.5r Multiply 2 * 3.14 414.48 = 6.28r2 + 6.28 * 5.5r Multiply 6.28 * 5.5 414.48 = 6.28r2 + 34.54r Reorder the terms: 414.48 = 34.54r + 6.28r2 Solving 414.48 = 34.54r + 6.28r2 Solving for variable 'r'. Reorder the terms: 414.48 + -34.54r + -6.28r2 = 34.54r + -34.54r + 6.28r2 + -6.28r2 Combine like terms: 34.54r + -34.54r = 0.00 414.48 + -34.54r + -6.28r2 = 0.00 + 6.28r2 + -6.28r2 414.48 + -34.54r + -6.28r2 = 6.28r2 + -6.28r2 Combine like terms: 6.28r2 + -6.28r2 = 0.00 414.48 + -34.54r + -6.28r2 = 0.00 Begin completing the square. Divide all terms by -6.28 the coefficient of the squared term: Divide each side by '-6.28'. -66 + 5.5r + r2 = 0 Move the constant term to the right: Add '66' to each side of the equation. -66 + 5.5r + 66 + r2 = 0 + 66 Reorder the terms: -66 + 66 + 5.5r + r2 = 0 + 66 Combine like terms: -66 + 66 = 0 0 + 5.5r + r2 = 0 + 66 5.5r + r2 = 0 + 66 Combine like terms: 0 + 66 = 66 5.5r + r2 = 66 The r term is 5.5r. Take half its coefficient (2.75). Square it (7.5625) and add it to both sides. Add '7.5625' to each side of the equation. 5.5r + 7.5625 + r2 = 66 + 7.5625 Reorder the terms: 7.5625 + 5.5r + r2 = 66 + 7.5625 Combine like terms: 66 + 7.5625 = 73.5625 7.5625 + 5.5r + r2 = 73.5625 Factor a perfect square on the left side: (r + 2.75)(r + 2.75) = 73.5625 Calculate the square root of the right side: 8.576858399 Break this problem into two subproblems by setting (r + 2.75) equal to 8.576858399 and -8.576858399.Subproblem 1
r + 2.75 = 8.576858399 Simplifying r + 2.75 = 8.576858399 Reorder the terms: 2.75 + r = 8.576858399 Solving 2.75 + r = 8.576858399 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-2.75' to each side of the equation. 2.75 + -2.75 + r = 8.576858399 + -2.75 Combine like terms: 2.75 + -2.75 = 0.00 0.00 + r = 8.576858399 + -2.75 r = 8.576858399 + -2.75 Combine like terms: 8.576858399 + -2.75 = 5.826858399 r = 5.826858399 Simplifying r = 5.826858399Subproblem 2
r + 2.75 = -8.576858399 Simplifying r + 2.75 = -8.576858399 Reorder the terms: 2.75 + r = -8.576858399 Solving 2.75 + r = -8.576858399 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-2.75' to each side of the equation. 2.75 + -2.75 + r = -8.576858399 + -2.75 Combine like terms: 2.75 + -2.75 = 0.00 0.00 + r = -8.576858399 + -2.75 r = -8.576858399 + -2.75 Combine like terms: -8.576858399 + -2.75 = -11.326858399 r = -11.326858399 Simplifying r = -11.326858399Solution
The solution to the problem is based on the solutions from the subproblems. r = {5.826858399, -11.326858399}
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